# idealizer -- compute Hom(I,I) as a quotient ring

## Synopsis

• Usage:
(F,G) = idealizer(I,f)
• Inputs:
• I, an ideal, whose endomorphism ring we'll compute
• f, , a nonzerodivisor in $I$
• Variable, ,
• Index, an integer,
• Optional inputs:
• Strategy => a list, default value {}, possible elements include Vasconcelos''
• Verbosity => an integer, default value 0, larger numbers give more information
• Index => ..., default value 0, Sets the starting index on the new variables used to build the endomorphism ring Hom(J,J)
• Variable => ..., default value w, Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).
• Outputs:
• F, , The inclusion map from $R$ into $S = Hom_R(I,I)$
• G, , $frac S \rightarrow frac R$, giving the fractions corresponding to each generator of $S$.

## Description

The idealizer of $I$, computed as target F, is the largest subring of the fraction field of ring I in which $I$ is still an ideal. Note that this is NOT the common use of the term in commutative algebra.

This is a key subroutine used in the computation of integral closures.

 i1 : R = QQ[x,y]/(y^3-x^7) o1 = R o1 : QuotientRing i2 : I = ideal(x^2,y^2) 2 2 o2 = ideal (x , y ) o2 : Ideal of R i3 : (F,G) = idealizer(I,x^2); i4 : target F QQ[w , x..y] 0,0 o4 = ------------------------------------- 2 2 2 3 5 (w x - y , w - x y, w y - x ) 0,0 0,0 0,0 o4 : QuotientRing i5 : first entries G.matrix 2 y o5 = {--, x, y} 2 x o5 : List